Diversification


BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Why diversify your investments?

  • What is the risk of one coin flip for $10?
    • The possible deviations from the mean are +10 and –10.
  • What is the risk of two coin flips for $5 each?
    • The possible deviations from the mean are
      • +10 (prob 1/4),
      • –10 (prob 1/4) and
      • 0 (prob 1/2).

More baskets is better

Variance and standard deviation

  • Variance is expected squared deviation from mean
    • Flipping once for $10 \(\Rightarrow\) variance = 100
    • Flipping twice for $5 \(\Rightarrow\) variance = (1/4) \(\times\) 100 + (1/4) \(\times\) 100 = 50
  • Standard deviation is square root of variance
    • Flipping once for $10 \(\Rightarrow\) std dev = 10
    • Flipping twice for $5 \(\Rightarrow\) std dev = 7.07

Portfolio Returns

  • Consider a $100,000 portfolio with 40% in one asset (asset 1) and 60% in a second (asset 2).
  • Suppose asset 1 \(\uparrow\) 20%. $40,000 \(\rightarrow\) $48,000
  • Suppose asset 2 \(\uparrow\) 10%. $60,000 \(\rightarrow\) $66,000
  • $100,000 \(\rightarrow\) $114,000
  • So, up 14% = 0.4 \(\times\) 20% + 0.6 \(\times\) 10%
  • The portfolio return is \(r_p = w_1r_1 + w_2r_2\) where \(w_i\) is the fraction of the portfolio invested in asset \(i\).

Expected portfolio return

  • \(\bar{r}_1\) = expected return of asset 1
  • \(\bar{r}_2\) = expected return of asset 2
  • The expected portfolio return is

\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]

Portfolio variance

  • The deviation of the portfolio return from its mean is

\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]

  • This equals

\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]

The deviation of the portfolio return from its mean is

\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]

and \((a+b)^2 = a^2 + b^2 + 2ab\), so the squared deviation is sum of

  • \(w_1^2(r_1-\bar{r}_1)^2\),
  • \(w_2^2(r_2 - \bar{r}_2)^2\),
  • \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\).

  • Expected value of \(w_1^2(r_1-\bar{r}_1)^2\) is \(w_1^2 \text{var}(r_1)\).
  • Expected value of \(w_2^2(r_2 - \bar{r}_2)^2\) is \(w_2^2 \text{var}(r_2)\).
  • Expected value of \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\) is \(2w_1w_2\text{cov}(r_1,r_2)\).
  • Portfolio variance is

\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]

2 Coin Flips Example

  • Suppose we start with $100
  • There are two independent assets, each with possible returns of \(\pm\) 10%
    • Put all $100 in a single asset \(\leftrightarrow\) flip once for $10
    • Put $50 in each asset \(\leftrightarrow\) flip twice for $5
  • std dev \(r_1 = 10\text{%}\) \(\Rightarrow\) var \(r_1 = 100\text{%}^2\)
  • std dev \(r_2 = 10\text{%}\) \(\Rightarrow\) var \(r_2 = 100\text{%}^2\)
  • assets are independent, so cov \(= 0\)

  • put $50 in each asset \(\Rightarrow\) weights = 1/2
  • portfolio variance is

\[\left(\frac{1}{2}\right)^2 100\text{%}^2 + \left(\frac{1}{2}\right)^2 100\text{%}^2 = 50\text{%}^2\]

  • portfolio std dev is \(\sqrt{50}\text{%}= 7.07\text{%}\)

Correlation and covariance

  • \(\text{cov} = \text{corr} \times \text{std}(r_1) \times \text{std}(r_2)\), so portfolio var is

\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2)\]

plus

\[2 w_1 w_2 \times \text{corr}(r_1, r_2) \times\text{std}(r_1)\times\text{std}(r_2)\]

Portfolio variance is lower when correlation is lower.

Correlation, covariance, and beta

The slope (beta) of the regression line of \(r_2\) on \(r_1\) is

\[\frac{\text{cov}(r_1, r_2)}{\text{var}(r_1)} = \text{corr}(r_1, r_2) \times \frac{\text{std}(r_2)}{\text{std}(r_1)}\]

Monthly returns of CVX and AAPL 2017–2021

std dev of CVX is  8.18%
std dev of AAPL is  8.50%
corr is  0.153
cov is  10.7%^2
beta is  0.16

Monthly returns of CVX and XOM 2017–2021

std dev of CVX is  8.18%
std dev of XOM is  8.58%
corr is  0.889
cov is  62.4%^2
beta is  0.93

CVX, AAPL, and XOM portfolios

  • The volatilities (std devs) of AAPL and XOM were very similar over the 2017-2021 period.
  • If you combined either with CVX, did you have similar volatilities over the period?
  • AAPL is a much better diversification play than XOM for someone holding CVX.

Some portfolios

cvx = 0.0818
aapl = 0.0850
xom = 0.0858
cvx_aapl = 0.153
cvx_xom = 0.889
w1 = 0.5
w2 = 0.5

# 50% CVX and 50% AAPL
w1**2 * cvx**2 + w2**2 * aapl**2 + 2*w1*w2*cvx_aapl*cvx*aapl

# 50% CVX and 50% XOM
w1**2 * cvx**2 + w2**2 * xom**2 + 2*w1*w2*cvx_xom*cvx*xom

Results

  • std dev of CVX with XOM is 8.14%
  • std dev of CVX with AAPL is 6.33%
  • compared to
    • std dev of CVX is 8.18%
    • std dev of XOM is 8.58%
    • std dev of AAPL is 8.50%